hodgkin huxley 07
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Nervous SystemNervous System
Signals are propagated from nerve cell toSignals are propagated from nerve cell tonerve cell (nerve cell (neuronneuron) via electro-chemical) via electro-chemicalmechanismsmechanisms
~100 billion neurons in a person~100 billion neurons in a person Hodgkin and Huxley experimented onHodgkin and Huxley experimented on
squids and discovered how the signal issquids and discovered how the signal isproduced within the neuronproduced within the neuron
H.-H. model was published inH.-H. model was published inJour. ofJour. ofPhysiologyPhysiology(1952)(1952)
H.-H. were awarded 1963 Nobel PrizeH.-H. were awarded 1963 Nobel Prize FitzHugh-Nagumo model is a simplificationFitzHugh-Nagumo model is a simplification
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Action PotentialAction Potential
Axonmembranepotential
difference V = VV = Vii V Vee
When the axonis excited, VV
spikes becausesodium Na+Na+and potassiumK+K+ ions flowthrough the
membrane.
10 msec
mV
_ 30
-70
V
_ 0
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C. George Boeree:www.ship.edu/~cgboeree/
NernstPotential
VVNaNa , VVKK and VVrr
Ion flow due to
electricalsignal
Travelingwave
http://www.ship.edu/~cgboeree/http://www.ship.edu/~cgboeree/http://www.ship.edu/~cgboeree/ -
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Circuit Model for AxonCircuit Model for Axon
MembraneMembraneSince the membrane separates charge, it ismodeled as a capacitor with capacitance CC. Ionchannels are resistors. 1/R = g =1/R = g = conductance
iiCC = C dV/dt= C dV/dt
iiNaNa = g= gNaNa (V (V
VVNaNa))
iiKK= g= gKK (V V(V VKK))
iirr = g= grr (V V(V Vrr))
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Circuit EquationsCircuit Equations
Since the sum of the currents is 0, it followsSince the sum of the currents is 0, it followsthatthat
aprrNa
IVVgVVgVVgdt
dVC KKNa += )()()(
where IIapap is applied current. If ion conductances are
constants then group constants to obtain 1st order,linear eq
apIVVgdtdVC += *)(
Solvinggives gIVtV ap /*)( +
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Variable ConductanceVariable Conductance
Experiments showed that gExperiments showed that gNaNa and gand gKK varied with timevaried with time
and V. After stimulus, Na responds much more rapidlyand V. After stimulus, Na responds much more rapidlythan K .than K .
g
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Hodgkin-Huxley SystemHodgkin-Huxley System
Four state variables are used:Four state variables are used:
v(t)=V(t)-Vv(t)=V(t)-Veqeq
is membraneis membrane
potential,potential,
m(t) is Na activation,m(t) is Na activation,
n(t) is K activation andn(t) is K activation and
h(t) is Na inactivation.h(t) is Na inactivation.
In terms of these variables ggKK=ggKKnn44 and
ggNaNa==ggNaNamm33hh.
The resting potential VVeqeq-70mV-70mV. Voltage clamp
experiments determined ggKK
and nn as functions of
tt and hence the parameter dependences on vv in
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Hodgkin-Huxley SystemHodgkin-Huxley System
IVvgVvngVvhmgdt
dvC aprrKNa
KNa+= )()()(
43
mvmvdt
dmmm )()1)(( =
nvnvdt
dnnn )()1)((
=
hvhv
dt
dhhh )()1)(( =
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IIapap =8, v(t)=8, v(t)
IIapap=7, v(t)=7, v(t)
m(t)
n(t)
h(t)
10msec
1.2
110 mV
40msec
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Fast-Slow DynamicsFast-Slow Dynamics
v, m are on a fast time scale and n, h arev, m are on a fast time scale and n, h are
slow.slow.
mm(v) dm/dt = m(v) dm/dt = m(v) (v) m.m.
mm(v) is much smaller(v) is much smaller
thanthannn(v) and(v) and hh(v). An(v). An
increase in v results inincrease in v results in
an increase in man increase in m(v)(v) andand
a large dm/dt. Hence Naa large dm/dt. Hence Naactivates more rapidlyactivates more rapidlythan K in response to athan K in response to a
change in v.change in v.
m(t)
n(t)
h(t)
10msec
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FitzHugh-Nagumo SystemFitzHugh-Nagumo System
Iwvfdt
dv+= )( wv
dt
dw5.0=and
II represents applied current, is small and f(v)f(v) is a cubicnonlinearity. Observe that in the (v,w)(v,w) phase plane
which is small unless the solution is near f(v)-w+f(v)-w+II=0=0. Thusthe slowmanifold is the cubic w=f(v)+w=f(v)+II whichwhich is the nullcline ofthe fast variable vv. And ww is the slow variable with nullcline
w=2vw=2v.
Iwvf
wv
dv
dw
+
= )(
)5.0(
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vv
I=0I=0 I=0.I=0.
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Stable rest
state
Stable
oscillation
Take f(v)=v(1-v)(v-a)f(v)=v(1-v)(v-a) .
w w
vv
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FitzHugh-Nagumo OrbitsFitzHugh-Nagumo Orbits
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ReferencesReferences1.1. C.G. Boeree,C.G. Boeree, The NeuronThe Neuron,, www.ship.edu/~cgboeree/.2.2. R. FitzHugh, Mathematical models of excitation andR. FitzHugh, Mathematical models of excitation and
propagation in nerve, In: Biological Engineering, Ed:propagation in nerve, In: Biological Engineering, Ed:H.P. Schwan, McGraw-Hill, New York, 1969.H.P. Schwan, McGraw-Hill, New York, 1969.
3.3. L. Edelstein-Kesket, Mathematical Models in Biology,L. Edelstein-Kesket, Mathematical Models in Biology,Random House, New York, 1988.Random House, New York, 1988.
4.4. A.L. Hodgkin, A.F. Huxley and B. Katz,A.L. Hodgkin, A.F. Huxley and B. Katz,J. PhysiologyJ. Physiology116116, 424-448,1952., 424-448,1952.
5.5. A.L. Hodgkin and A.F. Huxley,A.L. Hodgkin and A.F. Huxley,J. Physiol.J. Physiol.116,116, 449-566,449-566,1952.1952.
6.6. F.C. Hoppensteadt and C.S. Peskin,F.C. Hoppensteadt and C.S. Peskin, Modeling andModeling andSimulation in Medicine and the Life SciencesSimulation in Medicine and the Life Sciences, 2nd ed,, 2nd ed,Springer-Verlag, New York, 2002.Springer-Verlag, New York, 2002.
7.7. J. Keener and J. Sneyd,J. Keener and J. Sneyd, Mathematical PhysiologyMathematical Physiology,,Springer-Verlag, New York, 1998.Springer-Verlag, New York, 1998.
8.8. J. Rinzel,J. Rinzel, Bull. Math. BiologyBull. Math. Biology5252, 5-23, 1990., 5-23, 1990.
9.9.
E.K. Yeargers, R.W. Shonkwiler and J.V. Herod, AnE.K. Yeargers, R.W. Shonkwiler and J.V. Herod, AnIntroduction to the Mathematics of Biology: withIntroduction to the Mathematics of Biology: with